Zigzag Theorem for Partially Ordered Monoids

Abstract

While investigating the connection of absolute flatness of partially ordered monoids (briefly pomonoids) with their amalgamation, the author recently posed the question as to whether it was possible to formulate some ordered analogue of Isbell's zigzag theorem. Proving such a theorem is the primary aim of this article.

Epimorphisms, dominions and amalgamation in pomonoids

“…In fact these transformations come from those given in [2] p. 286; in [6] pp. 2573-2574 we observed that for pomonoids it suffices to consider rather the set F = {Ea, Eb, M, S, I a, I b, i} of above transformations.…”

“…We shall also revisit a zigzag theorem for pomonoids, proved in [6]. We begin by recalling dominions.…”

“…We shall mean by x −→ k y, x, y ∈ S 1 * U S 2 , that y is obtained from x by applying to it a transformation of type k ∈ F. The following lemmas are reproduced from [6] for ready reference.…”

“…One can further observe that this partial order renders the canonical map λ : S 1 * U S 2 −→ (S 1 * U S 2 )/λ monotone. In other words, λ is an order-congruence on S 1 * U S 2 , if we endow S 1 * U S 2 with the discrete order (for further details the reader is referred to [6], Sect. 2).…”

“…Definition 1 (Definition 1 of [6]). Let U be a subpomonoid of a pomonoid S. Then the subpomonoid dom S (U ) = {x ∈ S : for all pairs of pomonoid homomorphisms α, β : S −→ T with α | U = β | U , we have xα = xβ } is called the dominion of U (in S).…”